Determine the precise meaning of the inverse-Wigner R-transform

Determine the precise mathematical meaning of the analytically continued R-transform for the inverse of a Wigner matrix when ordinary inverse moments diverge and the standard Hermitian moment expansion is unavailable.

Background

The paper studies formal branches of the S-transform for the inverse of a Wigner matrix and obtains finite quantities that behave like regularized free cumulants. However, the inverse Wigner matrix has eigenvalues unbounded in the large-matrix limit because the Wigner spectrum reaches zero, so its ordinary inverse moments are singular. Consequently, the usual Hermitian construction of the R-transform does not apply, and the authors leave the interpretation of the analytically continued transform unresolved.

References

What remains to be understood is the precise meaning of this $R$-transform, which cannot be obtained from the standard Hermitian construction, but which nevertheless appears naturally through the analytic continuation of the non-Hermitian $S$-transform formalism.

— Spectral properties of deterministic matrices multiplied by rotationally invariant random non-Hermitian ensembles  (2609.03618 - Bousseyroux et al., 3 Sep 2026) in Section 5.5, "The inverse Wigner matrix"